The above inequality (REF ) is known as the Lieb–Thirring (LT) inequality. It was first established for \(\gamma =1\)  and \(n=3\)  by Lieb-Thirring [1]} as a key step toward proving the stability of matter in quantum mechanics. They subsequently extended the inequality in [2]}
to all \(n\ge 2\)  and \(\gamma >0\)  and for \(n=1\)  and \(\gamma >1/2\) . The case \(n=1\)  and \(\gamma =1/2\)  was settled by Weidl [3]}.
